Hamiltonian equations of motion of quadratic gravity

TL;DR

This paper derives and analyzes the Hamiltonian equations of motion for quadratic gravity, including linearized and cosmological solutions, highlighting conditions for consistency with general relativity.

Contribution

It provides explicit Hamiltonian equations for quadratic gravity and explores their linearized form and solutions in cosmological settings, using symbolic computation.

Findings

Linearized equations match covariant field equations under certain conditions.

Traceless spatial metric condition is necessary for Hamiltonian formulation validity.

Explicit solutions found for homogeneous and isotropic configurations.

Abstract

We compute explicitly the equations of motion of the Hamiltonian formulation of quadratic gravity. This is the theory with the most general Lagrangian with terms of quadratic order in the curvature tensor. We employ the symbolic computational tool Cadabra. We present the linearized version of the equations of motion, performing the longitudinal-transverse decomposition. We compare the linear equations with the covariant field equations, finding that, if general-relativity terms are active, the linear Hamiltonian formulation is valid only if the perturbative spatial metric is traceless, a condition that can be freely imposed by recurring to an arbitrary function. We apply the equations of motion on homogeneous and isotropic configurations, finding explicit solutions.